5 problems

Neighborhood reduction conjecture. Suppose the relations $\hat{\mathscr{B}}\cup\mathscr{C}$ span $\hat{\mathcal{R}}_*(w)$ for each neighborhood $J\subset S$ with $J=J^*$ and each…

Finite generalized half-braid spanning-set conjecture. There exists a finite subset $\mathscr{C}$ of the generalized half-braid relations for $(W,S,*)$ such that the relations…

Perfectly braided twisted Coxeter system conjecture. A twisted Coxeter system $(W,S,*)$ is perfectly braided if $s^*\neq s$ for all $s\in S$.

Let $S_n$ be the symmetric group, let $\mathcal{I}(S_n)$ denote its involutions, and let $\Gamma(x)$ and $\operatorname{Cyc}(y)$ be the sets associated with $x$ and $y$ in the pape…

Let $S_n$ be the symmetric group and let $\mathcal{I}(S_n)$ denote its involutions. For a fixed-point-free involution $y\in\mathcal{I}(S_n)$, let $\hat F^{\operatorname{FPF}}_y$ be…